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<h1 class="ltx_title ltx_title_section">Gaussian filter</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p"><a href="Convolution.html" title="" class="ltx_ref">Convolution</a> with a Gaussian kernel (also
referred to as a Gaussian blur or as a Gaussian filter) is one of
the most commonly used filters in image processing. The Gaussian kernel
is formed by a matrix that contains the values of a rotationally symmetric
Gaussian function</p>
<table id="Sx1.Ex1" class="ltx_equation">

<tr class="ltx_equation ltx_align_baseline">
<td class="ltx_eqn_pad"></td>
<td class="ltx_align_center"><img id="Sx1.Ex1.m1" class="ltx_Math" style="vertical-align:-19px" src="mi/mi61.png" width="281" height="47" alt="K_{\mathrm{G}}\left(x,y\mid\sigma\right)=a^{2}\exp{\left(-\frac{x^{2}+y^{2}}{2%
\sigma^{2}}\right)}\,."></td>
<td class="ltx_eqn_pad"></td>
</tr>
</table>
</div>
<div id="Sx1.p2" class="ltx_para">
<p class="ltx_p">Kernel <img id="Sx1.p2.m1" class="ltx_Math" style="vertical-align:-5px" src="mi/mi62.png" width="31" height="18" alt="K_{\mathrm{G}}"> of size <img id="Sx1.p2.m2" class="ltx_Math" style="vertical-align:-3px" src="mi/mi7.png" width="37" height="18" alt="l\times l"> is separable and the
vector <img id="Sx1.p2.m3" class="ltx_Math" style="vertical-align:-2px" src="mi/mi3.png" width="15" height="16" alt="\boldsymbol{k}"> (see <a href="Convolution.html" title="" class="ltx_ref">convolution with separable kernels</a>)
can be composed from values <img id="Sx1.p2.m4" class="ltx_Math" style="vertical-align:-14px" src="mi/mi65.png" width="198" height="35" alt="k_{i}\left(x\mid\sigma\right)=a\exp{\left(-\frac{x^{2}}{2\sigma^{2}}\right)}">,
where <img id="Sx1.p2.m5" class="ltx_Math" style="vertical-align:-5px" src="mi/mi4.png" width="87" height="19" alt="i=1,\ldots,l">, <img id="Sx1.p2.m6" class="ltx_Math" style="vertical-align:-6px" src="mi/mi35.png" width="135" height="21" alt="x=i-\left(l+1\right)/2">, <img id="Sx1.p2.m7" class="ltx_Math" style="vertical-align:-6px" src="mi/mi66.png" width="110" height="21" alt="l=1+2\left\lceil 3\sigma\right\rceil">,
and <img id="Sx1.p2.m8" class="ltx_Math" style="vertical-align:-2px" src="mi/mi23.png" width="13" height="12" alt="a"> is a scaling factor such that <img id="Sx1.p2.m9" class="ltx_Math" style="vertical-align:-7px" src="mi/mi64.png" width="79" height="22" alt="\sum_{i}k_{i}=1">. Users need
to input the standard deviation <img id="Sx1.p2.m10" class="ltx_Math" style="vertical-align:-2px" src="mi/mi63.png" width="15" height="12" alt="\sigma">.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Threshold for approximate localization of molecules</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">The <a href="../../detectors/Threshold.html" title="" class="ltx_ref">threshold</a> value can be
specified by users as an expression combining mathematical functions
and operators with variables based on the current raw or filtered
image. Variables provided by this filter are:</p>
</div>
<div id="Sx1.SSx1.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
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<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Gauss.I</span></td>
<td class="ltx_td ltx_align_left">current raw input image</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">Gauss.F</span></td>
<td class="ltx_td ltx_align_left">corresponding filtered image</td>
</tr>
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<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="Filters.html" title="" class="ltx_ref">Image filtering and feature enhancement</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="../../detectors/Detectors.html" title="" class="ltx_ref">Finding approximate positions of molecules</a></p>
</div>
</li>
</ul>
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